At aimple interest a sum become `6/5` of itsself in`2(1)/(2)` years. The rate of interest per annum is

Options

A. `6%`
B. `7.1/2%`
C. `8%`
D. `9%`

Answer

To find the rate of interest per annum when a sum becomes \( \frac{6}{5} \) of itself in \( 2 \frac{1}{2} \) years, we can follow these steps:

### Step 1: Understand the given information
We know that the amount becomes \( \frac{6}{5} \) of the principal (P) in \( 2 \frac{1}{2} \) years. This means:
\[ A = \frac{6}{5} P \]
Where \( A \) is the total amount after the interest is applied.

### Step 2: Calculate the simple interest (SI)
The simple interest can be calculated using the formula:
\[ SI = A - P \]
Substituting the values we have:
\[ SI = \frac{6}{5} P - P \]
To simplify this, we can express \( P \) as \( \frac{5}{5} P \):
\[ SI = \frac{6}{5} P - \frac{5}{5} P = \frac{1}{5} P \]

### Step 3: Use the formula for Simple Interest
The formula for simple interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) is the simple interest,
- \( P \) is the principal amount,
- \( R \) is the rate of interest per annum,
- \( T \) is the time in years.

### Step 4: Substitute the known values into the formula
We already calculated \( SI = \frac{1}{5} P \) and \( T = 2 \frac{1}{2} = \frac{5}{2} \) years. Now substituting these into the formula:
\[ \frac{1}{5} P = \frac{P \times R \times \frac{5}{2}}{100} \]

### Step 5: Cancel \( P \) from both sides
Assuming \( P \neq 0 \), we can divide both sides by \( P \):
\[ \frac{1}{5} = \frac{R \times \frac{5}{2}}{100} \]

### Step 6: Solve for \( R \)
Now, we can rearrange the equation to find \( R \):
\[ \frac{1}{5} = \frac{5R}{200} \]
Cross-multiplying gives:
\[ 200 \times 1 = 5R \times 5 \]
\[ 200 = 25R \]
Now, divide both sides by 25:
\[ R = \frac{200}{25} = 8 \]

### Step 7: Conclusion
The rate of interest per annum is:
\[ R = 8\% \]